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Last week, we had Peter Scholze give an interesting distinguished lecture series here at UCLA on “Prismatic Cohomology”, which is a new type of cohomology theory worked out by Scholze and Bhargav Bhatt. (Video of the talks will be available shortly; for now we have some notes taken by two note–takers in the audience on that web page.) My understanding of this (speaking as someone that is rather far removed from this area) is that it is progress towards the “motivic” dream of being able to define cohomology for varieties (or similar objects) defined over arbitrary commutative rings , and with coefficients in another arbitrary commutative ring . Currently, we have various flavours of cohomology that only work for certain types of domain rings and coefficient rings :

- Singular cohomology, which roughly speaking works when the domain ring is a characteristic zero field such as or , but can allow for arbitrary coefficients ;
- de Rham cohomology, which roughly speaking works as long as the coefficient ring is the same as the domain ring (or a homomorphic image thereof), as one can only talk about -valued differential forms if the underlying space is also defined over ;
- -adic cohomology, which is a remarkably powerful application of étale cohomology, but only works well when the coefficient ring is localised around a prime that is different from the characteristic of the domain ring ; and
- Crystalline cohomology, in which the domain ring is a field of some finite characteristic , but the coefficient ring can be a slight deformation of , such as the ring of Witt vectors of .

There are various relationships between the cohomology theories, for instance de Rham cohomology coincides with singular cohomology for smooth varieties in the limiting case . The following picture Scholze drew in his first lecture captures these sorts of relationships nicely:

The new prismatic cohomology of Bhatt and Scholze unifies many of these cohomologies in the “neighbourhood” of the point in the above diagram, in which the domain ring and the coefficient ring are both thought of as being “close to characteristic ” in some sense, so that the dilates of these rings is either zero, or “small”. For instance, the -adic ring is technically of characteristic , but is a “small” ideal of (it consists of those elements of of -adic valuation at most ), so one can think of as being “close to characteristic ” in some sense. Scholze drew a “zoomed in” version of the previous diagram to informally describe the types of rings for which prismatic cohomology is effective:

To define prismatic cohomology rings one needs a “prism”: a ring homomorphism from to equipped with a “Frobenius-like” endomorphism on obeying some axioms. By tuning these homomorphisms one can recover existing cohomology theories like crystalline or de Rham cohomology as special cases of prismatic cohomology. These specialisations are analogous to how a prism splits white light into various individual colours, giving rise to the terminology “prismatic”, and depicted by this further diagram of Scholze:

(And yes, Peter confirmed that he and Bhargav were inspired by the Dark Side of the Moon album cover in selecting the terminology.)

There was an abstract definition of prismatic cohomology (as being the essentially unique cohomology arising from prisms that obeyed certain natural axioms), but there was also a more concrete way to view them in terms of coordinates, as a “-deformation” of de Rham cohomology. Whereas in de Rham cohomology one worked with derivative operators that for instance applied to monomials by the usual formula

prismatic cohomology in coordinates can be computed using a “-derivative” operator that for instance applies to monomials by the formula

where

is the “-analogue” of (a polynomial in that equals in the limit ). (The -analogues become more complicated for more general forms than these.) In this more concrete setting, the fact that prismatic cohomology is independent of the choice of coordinates apparently becomes quite a non-trivial theorem.

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space , and is a vector in that Hilbert space, then one has

in the strong topology, where is the -invariant subspace of , and is the orthogonal projection to . (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if is a countable amenable group acting on a Hilbert space by unitary transformations , and is a vector in that Hilbert space, then one has

for any Følner sequence of , where is the -invariant subspace. Thus one can interpret as a certain average of elements of the orbit of .

I recently discovered that there is a simple variant of this ergodic theorem that holds even when the group is not amenable (or not discrete), using a more abstract notion of averaging:

Theorem 1 (Abstract ergodic theorem)Let be an arbitrary group acting unitarily on a Hilbert space , and let be a vector in . Then is the element in the closed convex hull of of minimal norm, and is also the unique element of in this closed convex hull.

*Proof:* As the closed convex hull of is closed, convex, and non-empty in a Hilbert space, it is a classical fact (see e.g. Proposition 1 of this previous post) that it has a unique element of minimal norm. If for some , then the midpoint of and would be in the closed convex hull and be of smaller norm, a contradiction; thus is -invariant. To finish the first claim, it suffices to show that is orthogonal to every element of . But if this were not the case for some such , we would have for all , and thus on taking convex hulls , a contradiction.

Finally, since is orthogonal to , the same is true for for any in the closed convex hull of , and this gives the second claim.

This result is due to Alaoglu and Birkhoff. It implies the amenable ergodic theorem (1); indeed, given any , Theorem 1 implies that there is a finite convex combination of shifts of which lies within (in the norm) to . By the triangle inequality, all the averages also lie within of , but by the Følner property this implies that the averages are eventually within (say) of , giving the claim.

It turns out to be possible to use Theorem 1 as a substitute for the mean ergodic theorem in a number of contexts, thus removing the need for an amenability hypothesis. Here is a basic application:

Corollary 2 (Relative orthogonality)Let be a group acting unitarily on a Hilbert space , and let be a -invariant closed subspace of . Then and are relatively orthogonal over their common subspace , that is to say the restrictions of and to the orthogonal complement of are orthogonal to each other.

*Proof:* By Theorem 1, we have for all , and the claim follows. (Thanks to Gergely Harcos for this short argument.)

Now we give a more advanced application of Theorem 1, to establish some “Mackey theory” over arbitrary groups . Define a *-system* to be a probability space together with a measure-preserving action of on ; this gives an action of on , which by abuse of notation we also call :

(In this post we follow the usual convention of defining the spaces by quotienting out by almost everywhere equivalence.) We say that a -system is *ergodic* if consists only of the constants.

(A technical point: the theory becomes slightly cleaner if we interpret our measure spaces abstractly (or “pointlessly“), removing the underlying space and quotienting by the -ideal of null sets, and considering maps such as only on this quotient -algebra (or on the associated von Neumann algebra or Hilbert space ). However, we will stick with the more traditional setting of classical probability spaces here to keep the notation familiar, but with the understanding that many of the statements below should be understood modulo null sets.)

A *factor* of a -system is another -system together with a *factor map* which commutes with the -action (thus for all ) and respects the measure in the sense that for all . For instance, the *-invariant factor* , formed by restricting to the invariant algebra , is a factor of . (This factor is the first factor in an important hierachy, the next element of which is the *Kronecker factor* , but we will not discuss higher elements of this hierarchy further here.) If is a factor of , we refer to as an *extension* of .

From Corollary 2 we have

Corollary 3 (Relative independence)Let be a -system for a group , and let be a factor of . Then and are relatively independent over their common factor , in the sense that the spaces and are relatively orthogonal over when all these spaces are embedded into .

This has a simple consequence regarding the product of two -systems and , in the case when the action is trivial:

Lemma 4If are two -systems, with the action of on trivial, then is isomorphic to in the obvious fashion.

This lemma is immediate for countable , since for a -invariant function , one can ensure that holds simultaneously for all outside of a null set, but is a little trickier for uncountable .

*Proof:* It is clear that is a factor of . To obtain the reverse inclusion, suppose that it fails, thus there is a non-zero which is orthogonal to . In particular, we have orthogonal to for any . Since lies in , we conclude from Corollary 3 (viewing as a factor of ) that is also orthogonal to . Since is an arbitrary element of , we conclude that is orthogonal to and in particular is orthogonal to itself, a contradiction. (Thanks to Gergely Harcos for this argument.)

Now we discuss the notion of a group extension.

Definition 5 (Group extension)Let be an arbitrary group, let be a -system, and let be a compact metrisable group. A-extensionof is an extension whose underlying space is (with the product of and the Borel -algebra on ), the factor map is , and the shift maps are given bywhere for each , is a measurable map (known as the

cocycleassociated to the -extension ).

An important special case of a -extension arises when the measure is the product of with the Haar measure on . In this case, also has a -action that commutes with the -action, making a -system. More generally, could be the product of with the Haar measure of some closed subgroup of , with taking values in ; then is now a system. In this latter case we will call *-uniform*.

If is a -extension of and is a measurable map, we can define the *gauge transform* of to be the -extension of whose measure is the pushforward of under the map , and whose cocycles for are given by the formula

It is easy to see that is a -extension that is isomorphic to as a -extension of ; we will refer to and as *equivalent* systems, and as *cohomologous* to . We then have the following fundamental result of Mackey and of Zimmer:

Theorem 6 (Mackey-Zimmer theorem)Let be an arbitrary group, let be an ergodic -system, and let be a compact metrisable group. Then every ergodic -extension of is equivalent to an -uniform extension of for some closed subgroup of .

This theorem is usually stated for amenable groups , but by using Theorem 1 (or more precisely, Corollary 3) the result is in fact also valid for arbitrary groups; we give the proof below the fold. (In the usual formulations of the theorem, and are also required to be Lebesgue spaces, or at least standard Borel, but again with our abstract approach here, such hypotheses will be unnecessary.) Among other things, this theorem plays an important role in the Furstenberg-Zimmer structural theory of measure-preserving systems (as well as subsequent refinements of this theory by Host and Kra); see this previous blog post for some relevant discussion. One can obtain similar descriptions of non-ergodic extensions by working relative to the invariant factor (or via the ergodic decomposition, if one has enough separability hypotheses on the system), but the result becomes more complicated to state, and we will not do so here; see this paper of Austin for details.

A dynamical system is a space X, together with an action of some group . [In practice, one often places topological or measure-theoretic structure on X or G, but this will not be relevant for the current discussion. In most applications, G is an abelian (additive) group such as the integers or the reals , but I prefer to use multiplicative notation here.] A useful notion in the subject is that of an (abelian) *cocycle*; this is a function taking values in an abelian group that obeys the *cocycle equation*

(1)

for all and . [Again, if one is placing topological or measure-theoretic structure on the system, one would want to be continuous or measurable, but we will ignore these issues.] The significance of cocycles in the subject is that they allow one to construct (abelian) *extensions* or *skew products* of the original dynamical system X, defined as the Cartesian product with the group action . (The cocycle equation (1) is needed to ensure that one indeed has a group action, and in particular that .) This turns out to be a useful means to build complex dynamical systems out of simpler ones. (For instance, one can build nilsystems by starting with a point and taking a finite number of abelian extensions of that point by a certain type of cocycle.)

A special type of cocycle is a *coboundary*; this is a cocycle that takes the form for some function . (Note that the cocycle equation (1) is automaticaly satisfied if is of this form.) An extension of a dynamical system by a coboundary can be conjugated to the trivial extension by the change of variables .

While every coboundary is a cocycle, the converse is not always true. (For instance, if X is a point, the only coboundary is the zero function, whereas a cocycle is essentially the same thing as a homomorphism from G to U, so in many cases there will be more cocycles than coboundaries. For a contrasting example, if X and G are finite (for simplicity) and G acts freely on X, it is not difficult to see that every cocycle is a coboundary.) One can measure the extent to which this converse fails by introducing the *first cohomology group* , where is the space of cocycles and is the space of coboundaries (note that both spaces are abelian groups). In my forthcoming paper with Vitaly Bergelson and Tamar Ziegler on the ergodic inverse Gowers conjecture (which should be available shortly), we make substantial use of some basic facts about this cohomology group (in the category of measure-preserving systems) that were established in a paper of Host and Kra.

The above terminology of cocycles, coboundaries, and cohomology groups of course comes from the theory of cohomology in algebraic topology. Comparing the formal definitions of cohomology groups in that theory with the ones given above, there is certainly quite a bit of similarity, but in the dynamical systems literature the precise connection does not seem to be heavily emphasised. The purpose of this post is to record the precise fashion in which dynamical systems cohomology is a special case of cochain complex cohomology from algebraic topology, and more specifically is analogous to singular cohomology (and can also be viewed as the group cohomology of the space of scalar-valued functions on X, when viewed as a G-module); this is not particularly difficult, but I found it an instructive exercise (especially given that my algebraic topology is extremely rusty), though perhaps this post is more for my own benefit that for anyone else.

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